# Constraint Equations

Constraint equations allow a node, in a particular direction to be constrained relative to a set of other nodes. Constraint equations are the fundamental building block for all other constraint types.

## Definition

**Name**

The name is only used as a convenient way of identifying a constraint equation.

**Stage List**

This specifies a stage list using one any of the forms detailed in “Lists”. If this is set to “all” the constraint equation applies to the whole model, irrespective of stage. Otherwise it applies only to the specified stage(s).

**Constrained**

The node coupled to a set of “primary” nodes so that it shares the coupled degrees of freedom

**Direction**

This is the direction in which the constrained node is constrained.

**Equation**

The equation is a set of factors, nodes and directions. The displacement of the “constrained” node is the summation of the factored displacements at the primary in the specified directions.

Any linear equation can be set up and there is no requirement that the constraint equation preserves equilibrium in model, so care should be taken that no artificial constraint is introduced into the model.

**Primary – Factor, Node and Direction**

The displacement of the constrained node is the summation of the factored displacements at the primary in the specified directions. Care should be taken when mixing translational and rotational freedom.

## Constraint Equations

Constraint equations are the most fundamental way of specifying a constraint on a node. This constrained node is related to one or more primary nodes. The displacement at the constrained node in a particular direction is a function of the displacements in a particular direction at a set of primary nodes. The basic relationship can be specified in the form:

So for a node whose displacement is the average of two other nodes this is of the form

For primary nodes 101 and 102 the equation would be of the form $0.5N_{101,x} + 0.5N_{102,x}$.

Rigid constraints and joints can be thought of as particular instances of constraint equations.

The axis directions of a constraint are the constraint axis directions of the constituent nodes. While a link element or rigid constraint uses the geometrical relationship between constrained and primary to ensure equilibrium, this is not true for the more general constraint equation. Thus care should be taken when defining constraint equations to ensure that no unwanted constraint is applied to the model. Also the factors in the constraint equation are dimensionless, so if these are to represent actual relative positions of nodes care should be taken with units.